High Quality Content by WIKIPEDIA articles! In mathematics, a quaternion algebra over a field F is a central simple algebra A over F[1][2] that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (=tensoring with a field extension), i.e. for a suitable field extension K of F, A otimes_F K is isomorphic to the 2x2 matrix algebra over K. The notion of a quaternion algebra can be seen as a generalization of the Hamilton quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over F = mathbb{R} (the real number field), and indeed the only one over R apart from the 2x2 real matrix algebra, up to isomorphism.

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